1) By Thm. 4.1 in Chap. III [1]}, \(z_-(t) \le z(t)\)  while both \(z_-(t)\)  and \(z(t)\)  remain in the domain \((-\infty , \overline{z})\) . It follows that \(z_-(t) \le z(t)\; \forall t \in \mathcal {T}\) . If \(z_+(\tau _-) \ge \overline{z}\) , then \(z_+(t) \ge \overline{z} > z(t) \; \forall t \in \mathcal {T}\)  by Lem. REF . In the remaining case where \(z_+(\tau _-) < \overline{z}\) , the inequality \(z_+(t) \ge z(t)\)  holds until \(z_+(t) <\overline{z}\)  by Thm. 4.1 in Chap. III [1]}. So if \(z_+(t)\)  does not arrive at \(\overline{z}\) , this inequality holds for all \(t \in \mathcal {T}\) . If \(z_+(t)\)  arrives at \(\overline{z}\)  at some time \(\tau \) , then \(z_+(t) \ge \overline{z} > z(t)\)  for \(t \in [\tau ,\tau _+]\)  by Lem. REF , whereas \(z_+(t) \ge z(t) \; \forall t \in [\tau _-,\tau ]\)  by the foregoing.
